Ab Initio Based Configuration Interaction Calculations on the Low

9876

J. Phys. Chem. A 1998, 102, 9876-9883

Ab Initio Based Configuration Interaction Calculations on the Low-Lying Electronic States of GaAs Biswabrata Manna and Kalyan Kumar Das* Department of Chemistry, Physical Chemistry Section, JadaVpur UniVersity, Calcutta 700 032, India ReceiVed: April 17, 1998; In Final Form: July 31, 1998

Multireference singles and doubles configuration interaction calculations which include relativistic effective core potentials have been performed on GaAs. Potential energy curves of 40 Λ-S states are computed. Spectroscopic constants (Te, re, and ωe) of 17 bound Λ-S states have been estimated and compared with the available observed and other calculated values. The lowest Λ-S state is ...π2 X3Σ-. All 22 Λ -S states which dissociate into Ga(2Pu) + As(4Su) and Ga(2Pu) + As(2Du) limits are taken into consideration for the spin-orbit CI calculations. Potential energy curves of 51 Ω-states are computed. The X3Σ0-+ component is found to be the ground state of GaAs in contrary to the experimental findings which suggest the other component X3Σ1- as the ground state. The splitting between the two components of X3Σ- is calculated to be 76 cm-1 as compared with the experimental value of 43 cm-1. The excited A3Π0+ state, which does not predissociate has re ) 2.757 Å, Te ) 22 178 cm-1, and ωe ) 125 cm-1. Te and ωe values are somewhat smaller, while re is larger than the observed data. Transition probabilities of several dipole allowed transitions are calculated. Three transitions such as 11Σ+-41Σ+, 21Σ+-41Σ+, and X3Σ--A3Π are found to be very strong. The radiative lifetimes of many excited states are also estimated. The A3Π0+ state at the vibrational level V′ ) 0 has the calculated radiative lifetime of 1050 ns which is somewhat larger than the observed range. 3Π-X3Σ-

I. Introduction Over the past decade there has been considerable interest in the experimental and theoretical studies of mixed group III-V and other semiconductor clusters and small molecules. Especially, compounds such as GaP, InP, GaAs, InSb and their neutral and ionic clusters of different sizes are important materials because of their technological applications. Extensive studies on the clusters of GaAs with different sizes have been made by Smalley and co-workers.1-5 Historically, O’Brien et al.1 have pioneered in generating the supersonic molecular beams of semiconductor clusters such as GaxAsy by the laser vaporization of pure GaAs crystal mounted on the side of a pulsed supersonic nozzle. These cluster-beams are characterized by laser photoionization followed by timeof-flight mass spectrometry measurements. Liu et al.2 have determined the electron affinities of GaAs, Si, and Ge clusters as a function of cluster size from the photodetachment and photofragmentation studies of these cluster anions. Photodissociations of the positive ions of these clusters are also studied.3 Wang et al.4 have studied the reactivity of NH3 on the gallium arsenide cluster surface. Ultraviolet photoelectron spectra of mass-selected negative gallium arsenide cluster ions have been recorded by Jin et al.5 Although clusters of gallium arsenide have been studied extensively, Lemire et al.6 are the first group who have studied the gas-phase diatomic GaAs molecules spectroscopically. The 3Π-X3Σ- electronic band system which is located in the range 23 000-24 800 cm-1 has been observed in a resonant twophoton ionization study of the jet-cooled GaAs molecule. The upper state, which is the third 3Π state, has been found to correlate with the Ga(2Pu) + As(2Du) dissociation limit. This * Author for correspondence.

band system is found to be useful in probing the reaction mechanism of organometallic vapor phase epitaxial growth of GaAs films on suitable substrates which provide a means for the production of electronic devices. High-resolution spectra of the 3Π-X3Σ- band taken by Lemire et al.6 have revealed that there occurs a predissociation of the 3Π state for V′ g 1 for Ω ) 2, 1, 0- components while the 0+ component survives. It is proposed that 2, 1, and 0- components of the repulsive 15Σ- state are coupled with the components of 3Π to induce the predissociation of the excited 3Π state. The lifetimes for individual vibronic levels of the excited 3Π state have been measured and found to be widely fluctuating ranging from τ ) 600 ns (V′ ) 0, Ω′ ) 0+) to τ ) 10 ns (V′ ) 1, Ω′ ) 2). The bond strength estimated from this experiment is found to be Do (GaAs) ) 2.06 ( 0.05 eV. Small gallium arsenide clusters are also studied by Van Zee et al.7 by using matrix-isolated ESR spectroscopy. Knight and Petty8 have observed ESR spectra of GaAs+ from the laser vaporization and photoionization of GaAs. Because of the technological applications of GaAs as a semiconducting material, several theoretical studies9-11 have been carried out for its bulk properties. Ab initio based CASSCF/CI calculations have been performed by Balasubramanian12-14 on GaAs, GaAs+, and GaAs- molecules. Meier et al.15 have carried out MRDCI calculations on GaAs and GaAs+ to compute potential energy curves and spectroscopic constants of several low-lying electronic states. However, none of these calculations include the spin-orbit interaction in the Hamiltonian. The transition probability of the observed transition 3Π-X3Σ- has not been calculated as yet. The electronic and geometrical structure of small GaxAsy clusters in stoichiometric and nonstoichiometric compositions have been calculated by using local spin density method.16,17 These calculations show

10.1021/jp981902d CCC: $15.00 © 1998 American Chemical Society Published on Web 11/10/1998

Calculations on Electronic States of GaAs

J. Phys. Chem. A, Vol. 102, No. 48, 1998 9877

that even numbered clusters tend to be singlets, while odd numbered belong to higher multiplets. The present paper deals with the computation of the potential energy curves and spectroscopic properties of many low-lying electronic states without and with spin-orbit coupling by using ab initio based multireference singles and doubles configuration interaction (MRDCI) method which takes care of the relativistic effects through the effective core potential. We have focused our attention on several dipole allowed transitions. The transition probabilities of these radiative transitions are calculated, and lifetimes of the upper states are also estimated and compared with observed data.

TABLE 1: Dissociation Correlation between the Atomic States and the Molecular States of GaAs without Including the Spin-Orbit Coupling

II. Method of Computations

spin components of the Λ-S eigenfunctions obtained from the above spin-independent calculations are included in the spinorbit treatment. For the even electron GaAs molecule, singlet, triplet, and quintet states of each symmetry are taken into consideration. A1, A2, and B1 representations consist of all spin-orbit states. Many-electron spin-orbit matrix elements for pairs of Ms values are computed. The spin-orbit CI wave functions are then used for getting the transition properties. Both Λ-S and Ω potential energy curves are fitted into polynomials, and vibrational Schro¨dinger equations are solved numerically.28 The electric transition dipole moments are averaged over pairs of vibrational functions involved in a given electronic transition. Transition probabilities of different dipole allowed transitions are computed. The radiative lifetimes of excited vibrational levels are obtained from their respective transition probabilities.

Relativistic effective core potentials (RECP) of the semicore type for both gallium and arsenic atoms are taken from Hurley et al.18 The 3d104s24p and 3d104s24p3 electrons for Ga and As, respectively, are being kept in the valence space, while the remaining inner electrons are replaced by RECPs. As a result, the number of active electrons for GaAs has been reduced to 28. The Gaussian basis sets chosen for the present calculations are of the type (3s3p4d/3s3p3d) which are optimized by Hurley et al.18 themselves. We have performed a self-consistent-field (SCF) calculation for a repulsive 15Σ- state with 28 valence electrons at each internuclear separation of the potential energy curve. The optimized SCF-MOs generated are used as one electron functions for the configuration interaction (CI) calculations. Preliminary calculations reveal that 20 electrons occupying 3d shells of Ga and As do not participate actively in forming the Ga-As bond in the low-lying excited states. We have, therefore, kept these d-electrons frozen in CI steps. Hence only eight electrons remain active in the CI space. The MRDCI method of Buenker and co-workers19-22 has been employed for the calculation of the ground and low-lying excited Λ -S states without the spin-orbit interaction but with the inclusion of all spin independent relativistic effects by using RECPs. The CI codes of Buenker and co-workers19-22 have been used for the present calculations. The codes take care of different open shell configurations efficiently by using the Table CI algorithm.23,24 For simplicity, the present calculations have been performed in the C2V subgroup of C∞V. A set of main reference configurations is chosen for the lowest eight roots of each irreducible representation of a given spin multiplicity to describe low-lying excited states. All single and double excitations have been carried out from these reference configurations. The configuration selection and energy extrapolation technique along with the Davidson correction25,26 are used to estimate the full CI energy in the same AO basis. The threshold value for the configuration selection is chosen as 2µ hartree throughout the calculation. The dimensions of the secular equations corresponding to the generated CI space are of the order of 400 000, while for the selected configuration space it is around 10 000. The CI wave functions obtained from these calculations are used to compute the one electron property matrix elements. 2 The spin-orbit calculations are carried out in the C2V group. The spin-orbit operators which are derived from RECPs of Ga and As are taken from Hurley et al.18 The eigenfunctions obtained from the spin independent CI are multiplied with appropriate spin functions which transform as C2V irreducible representations. The estimated full CI energies obtained in Λ-S CI calculations are kept as diagonal Hamiltonian matrix elements, while the off-diagonal elements are computed by using the RECP spin-operators and Λ-S CI wave functions. The Wigner-Eckart theorem is used to compute final results.27 All

molecular states (Λ-S) Σ , Π, 5Σ-, 5Π

atomic states in the dissociation limits Pu(Ga) + 4Su(As) 2 u(Ga) + Du(As)

0 10790

0 14353

Pu(Ga) + 2Pu(As)

18530

19774

3 - 3

2

1Σ+, 1Σ-(2), 1Π(3), 1∆(2),

2P

1Φ, 3Σ+, 3Σ-(2),

3Π(3), 3∆(2), 3Φ Σ (2), 1Σ-, 1Π(2), 1∆, 3 + Σ (2), 3Σ-, 3Π(2), 3∆

1 +

a

relative energies (in cm-1) expta calculation

2

Moore’s table [ref 29]; energies are averaged over j.

III. Computed Results and Discussion Potential Energy Curves and Composition of Λ-S States. The ground state (2Pu) of Ga combines with the ground state (4Su) of As to generate four states of triplet and quintet multiplicities: 3Σ-, 3Π, 5Σ-, and 5Π. However, the combination of the ground state of Ga with the low-lying 2Du and 2Pu excited states of As results a large number of molecular states of the singlet and triplet multiplicities. Table 1 shows the dissociation relationship between the atomic states and the molecular states of GaAs in the absence of any spin-orbit mixing. We have compared the relative energies at the dissociation limits calculated in this work with the experimental data29 averaged over j. The computed energies are found to be larger than the observed data by 1200-3500 cm-1. In the present study we have considered all possible 34 Λ-S states which dissociate into Ga(2Pu) + As(4Su), Ga(2Pu) + As(2Du), and Ga(2Pu) + As(2Pu) limits and six other states which correlate with still higher dissociation limits. In Figure 1a, we have plotted the computed potential energy curves of all triplet and quintet Λ-S states of GaAs, while singlet curves are drawn in Figure 1b. The corresponding electronic energy differences (Te), equilibrium bond distances (re), and vibrational frequencies (ωe) of 17 bound Λ-S states of GaAs are given in Table 2. The remaining 23 Λ-S states are repulsive in nature. The ground state of GaAs is X3Σ- with an open shell configuration. The calculated equilibrium bond length is about 0.14 Å larger than the observed re. However, the vibrational frequency is about 36 cm-1 smaller. The ground-state wave function is dominated by the open shell configuration ...π2 which generates two more states: 11∆ and 21Σ+. Both of these states are very strongly bound. The 11∆ state is lying fifth in the energy ordering with Te ) 8498 cm-1. In addition to ...π2, 3% of the...ππ configuration contributes in the 11∆ state. The re and ωe of 11∆ are comparable with those of the ground state. The 21Σ+ state which is next to 11∆ is

9878 J. Phys. Chem. A, Vol. 102, No. 48, 1998

Figure 1. (a) Potential energy curves of triplet and quintet Λ-S states of GaAs. (b) Potential energy curves of singlet Λ-S states of GaAs.

lying 13 956 cm-1 above the ground state. Two more configurations: ...π4 and ...π3π contribute significantly to the description of the 21Σ+ state. The Ga-As bond in the 21Σ+ state is found to be relatively strong with re ) 2.515 Å and ωe ) 269 cm-1. The X3Σ-, 11∆, and 21Σ+ states dissociate into three different limits in the increasing order. The lowest excited 13Π state arises from a single excitation 2 2 σ3π f σ3π3 which also generates the corresponding singlet counterpart. The 13Π state is only 1066 cm-1 above the ground state, while the transition energy of 11Π is about 5941 cm-1. The equilibrium bond lengths of both 13Π and 11Π are comparatively shorter than the ground-state re. The vibrational frequencies of these two states are in the range 235-270 cm-1. The 13Π state correlates with Ga(2Pu) + As(4Su), while the 11Π state dissociates into the upper limit Ga(2Pu) + As(2Du). Besides ...σ3π3, there exists one more configuration ...σ3π2π which contributes significantly to both the low-lying Π states. The fourth Λ-S state of GaAs is a stable 11Σ+ state which is

Manna and Das dominated by a closed-shell configuration ...π4 (c2 ) 0.52) and three open shell configurations: ...π3π (c2 ) 0.28), ...σ2σ3π4 (c2 ) 0.06), and ...π2 (c2 ) 0.02). The 11Σ+ state has the shortest bond length (re ) 2.21 Å) and comparatively large vibrational frequency (ωe ) 294 cm-1), and it dissociates into the second asymptote Ga(2Pu) + As(2Du). The 23Π state is relatively weakly bound with a broad and shallow potential well. Single excitations of the type σ f π* and σ f π from the ground-state configuration ...σ2π2 dominate in the formation of the 23Π state. It is noted that the 13Π state is predominantly ...σ3π3. The ...σπ2π configuration gives rise to 10 Λ-S states: 1Π (3), 1Φ, 3Πr (2), 3Πi (2), 3Φ, and 5Π. Of these, both 11Φ and 13Φ states are repulsive in nature and they dissociate into the second asymptote, while the repulsive 15Π state correlates with the ground dissociation limit. Among three 1Π states originating from ...σπ2π, only 21Π is very weakly bound with ωe ) 98 cm-1, while 31Π and 41Π are repulsive. The third 3Π state which is designated here as A3Π has a special importance because of the observed A3Π-X3Σ- band system.6 The remaining two 3Π states, namely, 43Π and 53Π, are repulsive. Analyzing the CI wave functions it has been found that the A3Π state has about 81% character of the ...σ3π2π configuration. The computed transition energy of A3Π is 22 250 cm-1 which matches well with the observed average Te of 23 545 cm-1. The observed re and ωe values reported in Table 2 are for the 0+ component of A3Π for which other Ω components are found to predissociate. Thus the inclusion of the spin-orbit interaction is desired for the meaningful comparison with the observed data. We shall return to this aspect in the next section where we discuss the effect of the spinorbit coupling. There is a strongly bound 13Σ+ state with a Te value of 21 852 cm-1. This state arises mainly from the closed-shell 11Σ+ state through a single excitation π4 f π3π*. However, several other multiple excitations also occur with smaller contributions. The 23Σ- state has a very shallow potential well with much longer bond length. Figure 1a shows an avoided crossing of this state with its next higher repulsive state 33Σ- at around 5.3ao. Both these states dissociate into the second asymptote. The 23Σ+ state comes next in the energy ordering. It is seen from the adiabatic curve in Figure 1a that the 23Σ+ curve undergoes strong avoided crossings with the weakly bound and repulsive potential energy curves of 33Σ+ and 43Σ+ at around 4.7ao and 5.7ao, respectively. As a result the adiabatic curve of the 23Σ+ state shows a double minima. The minimum appearing at the long distance is very shallow. The minimum at the shorter distance (re ) 2.263 Å) lying 33 270 cm-1 above the ground state is very deep and its composition at re is found to be a mixture of ...σ2σ3π4 (c2 ) 0.4), ...σ2σ3π3π (c2 ) 0.27), and ...σ22π3π (c2 ) 0.19). The computed CI wave functions show similar mixing for the 33Σ+ state at r ) 5.2ao and for the 43Σ+ state at r ) 6.5ao. This essentially confirms strong avoided crossings among these states. However, the fitted potential well of 23Σ+ with shorter re contains only three vibrational levels. A strongly bound excited 41Σ+ state has been obtained at 33 776 cm-1 of energy in our calculations. This state has a minimum at 2.586 Å with ωe ) 172 cm-1. A sharp avoided crossing at 5.2ao between the bound 41Σ+ state and a repulsive 31Σ+ state is seen in Figure 1b. The diabatic potential energy curve of the 41Σ+ state dissociates into the higher asymptote. At around the equilibrium bond length, the 41Σ+ state shows complicated character, and six configurations interact strongly. However, to adequately describe so highly excited states



TABLE 2: Spectroscopic Parameters of the Bound Λ-S States of GaAs Te (cm-1)

state X3 Σ

13Π 11Π 11Σ+ 11∆ 21Σ+ 23Π 13Σ+ 23ΣA3Π 21Π 23Σ+ 41Σ+ 15Σ+ 25Π 15∆ 25Σ+

0 1066 (1830)b 5941 (6440)b 6690 (7768)b 8498 (7874)b 13956 (14383)b 16832 (18590)b (17700)d 21852 (23403)b 22020 22250 (24600)b (21600)d [23545]a 23953 (24679)c 33270 (31106)c 33776 35717 38843 (40352)c 43986 (45431)c 45157

ωe (cm-1)

re (Å) b

179 (215)b [215]a 236 (236)b 270 (277)b 294 (279)b 193 (214)b 269 (321)b 126 (135)b (132)d 209 (208)b 85 135 (160)b (150)d [152]a 98 (148)c 298 (201)c 172 178 138 (168)c 223 (287)c 196

2.670 (2.60) [2.53]a 2.398 (2.38)b 2.356 (2.34)b 2.210 (2.23)b 2.651 (2.58)b 2.515 (2.47)b 3.207 (3.10)b (3.21)d 2.374 (2.405)b 3.766 2.717 (2.68)b (2.70)d [2.662]a 2.789 (2.693)c 2.263 (2.421)c 2.586 2.546 2.957 (2.818)c 2.460 (2.449)c 2.616

a Experimental values [ref 6]. b CASSCF/SOCI calculated results [ref 14]. c CASSCF/FOCI calculated results [ref 14]. d Other calculated results [ref 15].

especially, near the dissociation limit, it would be necessary to use Rydberg functions.15 Among the quintet states studied here, 15Π and 15Σ- are repulsive and dissociate into the ground atomic states Ga(2Pu) + As(4Su). The 15Π state is mainly dominated by the ...σ3π2π configuration, while the 15Σ- state is characterized by the ...σ3σ4π2 (c2 ) 0.88) configuration. The remaining four quintet states of GaAs, 15Σ+, 25Π, 15∆, and 25Σ+, are high-lying but strongly bound (see Figure 1a). The 15Σ+ and 25Σ+ states are separated by about 9440 cm-1. The 15Σ+ state has a shorter bond length (re ) 2.546 Å) than 25Σ+ (re ) 2.616 Å), while the vibrational frequency of the former state is lower by about 20 cm-1 (see Table 2). Three configurations: ...σ2σ3π3π, ...σ22π2π2, and ...σ2σ3π2π2 contribute significantly in the characterization of both these states. Of the remaining two quintet states, 15∆ is lying 43 986 cm-1 above the ground state, and it has a shorter re (2.46 Å) with ωe ) 223 cm-1. This state has almost a pure character of ...σ2σ3π3π (c2 ) 0.92). The 25Π state which is characterized dominantly by the ...σ2σ32π2π configuration has a longer re and smaller ωe. These highly excited quintet states require Rydberg functions in the basis set for their adequate description.15 The present calculations do not include any Rydberg functions, and hence these quintets may not be very accurate. The ground-state dissociation energy of GaAs estimated from the MRDCI calculation has been found to be 1.3 eV which is much lower than the value 2.06 ( 0.05 eV obtained from the experimental data.6 The present value is comparable with the CASSCF/FOCI De value.14 Meier 15 has shown an improvement of about 0.2 eV due to the d-correlation. The extension of the basis set would also improve the De value as predicted by Balasubramanian.14 Effects of the Spin-Orbit Coupling. Spin-orbit coupling is introduced through the inclusion of the spin-orbit operator18 derived from RECPs. As a result Λ-S states of different symmetries interact through the same Ω-components. In Table 3, we have given the dissociation correlation between the molecular Ω states and the atomic states in the presence of the spin-orbit interaction. Here in this study we have considered only those Λ-S states which dissociate into the lowest two asymptotes: 2Pu(Ga) + 4Su(As) and 2Pu(Ga) + 2Du(As). In otherwords, we have included 22 Λ-S states in the spin-orbit CI calculations. Due to the spin-orbit coupling, two dissocia-

TABLE 3: Dissociation Correlation between the Atomic States and the Molecular States of GaAs with the Inclusion of the Spin-Orbit Coupling

molecular states (Ω) 2,1,1,0+,03,2,2,1,1,1,0+,0+,0-,02,1,1,0+,03,2,2,1,1,0+,03,2,2,1,1,1,0+,0+,0-,04,3,3,2,2,2,1,1,1,1,0+,0+,0-,0aMoore’s

atomic states in relative energies (in cm-1) the dissociation limits Ga + As expt.a calculation 2P 1/2 2P 3/2 2P 1/2 2P 1/2 2P 3/2 2P 3/2

+ 4S3/2 + 4S3/2 + 2D3/2 + 2D5/2 + 2D3/2 + 2D5/2

0 826 10593 10915 11419 11741

0 735 14129 14526 14884 15266

table [ref 29].

tion limits split into six asymptotes. Relative energies of these asymptotes are compared with the experimental data.29 The computed energies are somewhat larger than the observed data except for the lowest two limits for which the agreement is good. There are 51 spin-orbit states with Ω ) 0+, 0-, 1, 2, 3, and 4 generated from 22 Λ-S states. In Figure 2, potential energy curves of nine states with Ω ) 0+ are plotted. The X3Σ0-+ component dissociates into Ga(2P1/2) + As(4S3/2) while the 13Π0+ and 15Π0+ components correlate with Ga(2P3/2) + As(4S3/2). Several sharp avoided crossings are noted in Figure 2. Potential energy curves of all nine states of the 0- symmetry are shown in Figure 3. The 0- component of 13Π correlates with the lowest asymptote. We have plotted potential energies of the states with Ω ) 2 and 4 in Figure 4. The remaining 16 states with Ω ) 1 and two states with Ω ) 3 are drawn in Figure 5. We have left out three repulsive states with Ω ) 3 to avoid further complicacy in Figure 5. Spectroscopic constants of 18 low-lying Ω states below 23 000 cm-1 are reported in Table 3. The ground state splits into two components: X3Σ0-+ and X3Σ1-. The present calculations predict the X3Σ0-+ component to be lower than the other by about 76 cm-1. However, highresolution spectroscopic experiments on jet-cooled GaAs by Lemire et al.6 have shown the opposite trend. The X3Σ1- state has been identified as the ground state with the X3Σ0-+ substate lying 43 cm-1 above it. The CI estimated equilibrium bond lengths of both components of the X3Σ- state are about 0.12 Å longer than the observed values. The corresponding calculated ωe values are smaller than the experimental results by about 30 cm-1. Four components of the 13Π state are separated at the


Figure 2. Potential energy curves of low-lying Ω ) 0+ states of GaAs.

Manna and Das

Figure 4. Potential energy curves of low-lying Ω ) 2 and 4 states of GaAs (the curve with dashed line is for Ω ) 4).

Figure 3. Potential energy curves of low-lying Ω ) 0- states of GaAs.

most by about 1050 cm-1. The 13Π2 and 13Π0- substates do not have any nearby components of the same symmetries for mixing. As a result, these two components remain almost pure 13Π (see Table 4). Two other components 13Π0+ and 13Π1 interact with X3Σ0+ and X3Σ1 , respectively. As seen from 3 Table 4, about 67% 1 Π and 30% 13Σ- contribute to the 13Π0+ state at re. Multiple avoided crossings are also evident from Figures 2 and 5. The 11Π1 component is not affected much by the spin-orbit coupling. Its Te value is increased by 350 cm-1. A similar situation has been noted for three successive singlet states: 11Σ0++, 11∆2, and 21Σ0++. The energies of the four components of the next important 23Π state are 2, 1, 0+, and 0- in the ascending order. Although these Ω-states undergo avoided crossing, we have fitted the diabatic potential energy curves for spectroscopic parameters listed in Table 4. Three components Ω ) 2, 1, and 0- of the repulsive 15Σ- state undergo avoided crossings with the same components of the 23Π state. Both the components of the 13Σ+ state show multiple

Figure 5. Potential energy curves of low-lying Ω ) 1 and 3 states of GaAs (curves with dashed lines are for Ω ) 3).

crossings within a small range of bond distance in the potential energy curves. We could not fit the diabatic potential energies of the 13Σ0+- and 13Σ+ 1 components. The adiabatic curves are found to be predissociating. However, there are no experimental evidences for these states. The third 3Π state which we designate as A3Π is spectroscopically observed. The A3Π-X3Σ- band system in the jetcooled GaAs molecule has been studied in detail.6 The present spin-orbit CI calculations reveal four components A3Σ0-, A3Π0+, A3Π1, and A3Π2 in the same energy ordering as the experimental results. Transition energies of these components compare well with the observed values. Table 4 shows that the composition of these Ω-states is mainly A3Π. Figure 2 shows that the A3Π0+ component survives the predissociation, while the remaining three components A3Π0-, A3Π1, and A3Π2 predissociate mainly due to their interactions with the corre-



TABLE 4: Spectroscopic Parameters of Low-Lying Spin-Orbit States of GaAsa state

Te (cm-1)

re (Å)

ωe (cm-1)

dominant Λ-S states at reb

X3∑0-+ X3Σ1 13Π2 3 1 Π1 13Π0+ 13Π011Π1 11∑0++ 11∆2 21∑0++ 23Π2 23Π1 23Π023Π0+ A3Π0A3Π0+ A3Π1 A3Π2

0 [43] 76 [0] 736 1215 1751 1783 6283 6947 8771 14231 16794 17047 17203 17327 22142 [23545] 22178 [23568] 22265 [23760] 22650 [24079]

2.673 [2.55] 2.667 [2.53] 2.396 2.388 2.389 2.401 2.356 2.218 2.649 2.523 3.218 3.200 3.172 3.176 2.730 2.757 [2.662] 2.716 2.694

180 [209] 181 [215] 234 240 234 231 265 297 189 270 122 120 128 135 143 125 [152] 145 135

X3Σ- (97) X3Σ- (97), 13Π (2.1) 13Π (99.8) 13Π (97), 13Σ- (2.3) 13Π (67), 13Σ- (30.2) 13Π (99.8) 11Π (98) 11Σ+ (99) 11∆ (99) 21Σ+ (99) 23Π (79), 15Σ- (19.5) 23Π (56), 15Σ- (41.6) 23Π (83.6), 25Σ- (14.7) 23Π (98) A3Π (98) A3Π (99) A3Π (94), 21Π (5) A3Π (99)

Figure 7. Transition dipole moment functions for 0+-0+ transitions of the A3Π0+ state.

a Values in square brackets represent the experimental values [ref 6]. b Values in parentheses are percentage contributions.

Figure 8. Transition dipole moment functions for 0+-1 transitions of the A3Π0+ state.

TABLE 5: Radiative Lifetimes of Some Excited Λ-S States (W′ ) 0) of GaAs

Figure 6. Transition dipole moment functions for A3Π-X3Σ- and A3Π-13Π transitions.

sponding Ω-components of the repulsive 15Σ- state as evident from Figures 2-5. This feature confirms the experimental findings of Lemire et al.6 Transition Moments, Transition Probabilities, and Radiative Lifetimes. Experimentally,6 vibronic bands of the A3ΠX3Σ- system of GaAs have been studied by using highresolution spectroscopic measurement. The measured lifetimes of individual vibronic levels of the A3Π excited state of GaAs fluctuate very widely as functions of V′, Ω′, and isotopic composition. It falls in the range 600 ns-10 ns. In the present study we have calculated transition probabilities of several transitions which are allowed in Hund’s case (a). In addition, we have also considered all possible transitions of type 0+-0+ and 0+-1 which become allowed in case (c). Figure 6 shows transition dipole moments of two important transitions A3Π-X3Σ- and A3Π-13Π as a function of the internuclear distance. Both the functions are smooth having maxima. The transition moment of the former transition is generally higher than that of the later. The transition dipole moments of 0+-0+ and 0+-1 type transitions with the upper state being the component of A3Π are plotted in Figures 7 and 8, respectively. There are four allowed 0+-0+ transitions, namely, A3Π0+-13Π0+, A3Π0+-11∑0++, A3Π0+-X3∑-0+, and A3Π0+-21∑0++, while three perpendicular 0+-1 transitions 3 3 3 + + (∆Ω ) (1) are A3Π0+-X3Σ1 , A Π0 -1 Π1, and A Π0 11Π1. We have estimated the transition probabilities from the

transition

partial lifetime of the upper state at V′ ) 0 (in sec)

11Π r 21Σ+ 11Σ+ r 21Σ+ 13Π r 13Σ+ 11Σ+ r 41Σ+ 21Σ+ r 41Σ+ 11Π r 41Σ+ X3Σ- r 23Π 13Π r 23Π X3Σ- r A3Π 13Π r A3Π 13Σ+r 23Σ+ 13Π r 23Σ+

8.04 × 10-6 46.8 × 10-6 1.3 × 10-6 48.3 × 10-9 396.5 × 10-9 6.8 × 10-4 82.8 × 10-6 1.3 × 10-1 456 × 10-9 14.8 × 10-6 19.1 × 10-6 2.5 × 10-6

total lifetime of the upper state at V′ ) 0 (in sec) τ (21Σ+) ) 6.9 × 10-6 τ (13Σ+) ) 1.3 × 10-6 τ (41Σ+) ) 43.1 × 10-9 τ (23Π) ) 82.7 × 10-6 τ (A3Π) ) 440 × 10-9 τ (23Σ+) ) 2.2 × 10-6

Einstein spontaneous emission coeffecients AV′V′′ (s-1) between various vibrational levels V′ of the upper electronic state and V′′ level of the lower state. The calculations desire the knowledge of the electronic transition moment function which are taken from the curves shown in Figures 6-8. The radiative lifetimes are basically the reciprocal of the sum of AV′V′′ over V′′ for a given V′ level of the upper state. Table 5 displays the estimated radiative lifetimes of the upper electronic Λ-S states at V′ ) 0 only for several dipole allowed transitions. Our calculations suggest that 11Σ+-41Σ+ is the strongest possible transition with the shortest partial lifetime, although it is not experimentally observed so far. The second allowed transition from the 41Σ+ state, namely, 21Σ+-41Σ+, has also large transition probability. The partial lifetime of the experimentally observed transition X3Σ--A3Π is about 456 ns. However, this estimation does not take into account of the spinorbit interaction. The total lifetime of the 41Σ+ state at V′ ) 0 is found to be 43 ns, which is the shortest-lived state among


Manna and Das

TABLE 6: Radiative Lifetimes of the A3Π0+ State of GaAs lifetimes of the A Π0 state (µs) 3

+

transition

V′ ) 0

V′ ) 1

V′ ) 2

3 + X3∑0+ r A Π0 13Π0+ r A3Π0+ 3 + 21∑+ 0+ r A Π0 3Π + 11∑+ r A 0 0+ 3 + X3Σ1 r A Π0 3 1 Π1 r A3Π0+ 11Π1 r A3Π0+ total lifetimes τ (A3Π0+)

69.4 10.7 1.5 × 104 1.1 × 104 1.2 83.7 1.9 × 103

73.5 8.1 3.5 × 103 1.5 × 104 1.3 76.2 2.0 × 103

72.6 7.2 3.3 × 103 2.3 × 104 1.7 64.2 1.6 × 103

1.05

1.09

1.32

all. The observed A3Π state has a lifetime of about 440 ns, when no spin-orbit coupling is considered. In Table 6 we have given the radiative lifetimes of the 0+ component of A3Π for three vibrational levels V′ ) 0, 1, and 2. As per observation of Lemire et al.,6 other components of A3Π with Ω ) 2, 1, 0undergo predissociation. Summing up the transition probabilities of seven possible transitions from the A3Π0+ component (four 0+-0+ (∆Ω ) 0) and three 1-0+ (∆Ω ) ( 1)), the total lifetime estimated for the A3Π0+ state at V′ ) 0 is about 1050 ns and it increases with the increase in V′. The value is somewhat larger than the observed one which fluctuates widely between 600 and 10 ns. Comparison with Other Calculations. Several calculations on the neutral GaAs molecule and its ions GaAs+ and GaAshave been performed by Balasubramanian.14 These authors have computed spectroscopic constants and potential energy curves of Λ-S states up to the energy of 25 000 cm-1 by the complete active space MCSCF (CASSCF) followed by first-order CI (FOCI) and second-order CI (SOCI) calculations. Table 2 shows the comparison of the CASSCF/SOCI spectroscopic constants with our values. In general, re values calculated in this paper are somewhat larger, while the vibrational frequencies and the transition energies are smaller than Balasubramanian’s results. Three quintet states have been calculated by Balasubramanian using the CASSCF/FOCI method. We have not studied 5Π (III) which lies above 50 000 cm-1. However, some other strongly bound states such as 41Σ+, 15Σ+, and 25Σ+ lying below 50 000 cm-1 are not studied before. Large discrepancies may be noted for re and ωe values of 21Σ+ and 23Σ+ states. The strong avoided crossing in the potential energy curve of the 23Σ+ state is not shown in ref 14. The state which undergoes a sharp avoided crossing with a shallow potential energy curve of 31Σ+ is also not calculated in the work of Balasubramanian. Meier et al.15 have studied several excited states of GaAs by MRD-CI study. Three excited 3Π states are calculated by these authors. The 23Π state has been found to lie at 17 700 cm-1 compared with our Te values of 16 832 cm-1. The bond length is comparable, while the vibrational frequency is larger than our value by 6 cm-1. Spectroscopic features of all important A3Π state studied by Meier et al.15 matches well with the present study with the exception of ωe. None of the previous calculations has included the spin-orbit coupling. Therefore, the present work reports for the first time the effects of the spinorbit interaction. The predissociation of A3Π0-, A3Π1, and A3Π2 states through the coupling with the components of the repulsive 15Σ- state observed by Lemire et al.6 has also been suggested by Balasubramanian14 and Meier et al.15 The fact that the A3Π0+ component survives the predissociation has been nicely demonstrated in the potential energy curves of the Ω-states computed here.

IV. Conclusion MRDCI calculations of the GaAs molecule based on RECPs in which 3d, 4s, and 4p electrons are kept in the valence space confirm the ground state as X3Σ-. There are at least 17 bound Λ-S states within 45 000 cm-1 of energy. Spectroscopic parameters of the ground and third 3Π states designated as X3Σand A3Π, respectively, which are observed in the resonant two photon ionization spectroscopy by Lemire et al.6 compare well with our calculated results. The vibrational frequencies of some of the states of GaAs are found to be lower than the other available experimental or calculated data. The present calculations suggest that the splitting of the two spin-orbit components of the X3Σ- state is about 76 cm-1 with X3∑0-+ being the ground state. This is in contrary to the experimental findings of Lemire et al.6 according to which the ground state is X3Σ1, and the corresponding 0+ component is 43 cm-1 above it. Potential energy curves of four Ω-components of the A3Π state confirm the observed predissociation of the A3Π2, A3Π1, and A3Π0- states through the interaction with the corresponding components of the repulsive 15Σ- state. The A3Π0+ state survives from the predissociation. The calculated transition energy of A3Π0+ is about 22 178 cm-1 compared with the experimental value of 23 568 cm-1. The re value estimated in this calculation is 0.095 Å higher than the observed value of 2.662 Å, while the equilibrium vibrational frequency is lower than the experimental value by about 27 cm-1. Transition probabilities calculated from the CI wave functions show that the transition from the excited 41Σ+ state to the lowest state of the same symmetry is the strongest among all. The X3Σ-A3Π transition which is the only observed transition is also reasonably strong. The radiative lifetime of the A3Π state at V′ ) 0 is found to be 440 ns. The present calculations reveal that the total lifetimes of the A3Π0+ component at the lower vibrational levels V′ ) 0, 1, and 2 are 1050, 1090, and 1320 ns, respectively, as compared with the observed values in the range 10-600 ns. Acknowledgment. The authors thank Prof. R. J. Buenker, Wuppertal, Germany for giving us the permission to use his MRDCI codes. The financial support from the CSIR, India under the Grant 01(1427)/96/EMR-II is gratefully acknowledged. References and Notes (1) O’Brien, S. C.; Liu, Y.; Zhang, Q.; Heath, J. R.; Tittel, F. K.; Curl, R. F.; Smalley, R. E. J. Chem. Phys. 1986, 84, 4074. (2) Liu, Y.; Zhang, Q.; Tittel, F. K.; Curl, R. F.; Smalley, R. E. J. Chem. Phys. 1986, 85, 7434. (3) Zhang, Q.; Liu, Y.; Curl, R. F.; Tittel, F. K.; Smalley, R. E. J. Chem. Phys. 1988, 88, 1670. (4) Wang, L.; Chibante, L. P. F.; Tittel, F. K.; Curl, R. F.; Smalley, R. E. Chem. Phys. Lett. 1990, 172, 335. (5) Jin, C.; Taylor, K.; Conciecao, J.; Smalley, R. E. Chem. Phys. Lett. 1990, 175, 17. (6) Lemire, G. W.; Bishea, G. A.; Heidecke, S. A.; Morse, M. D. J. Chem. Phys. 1990, 92, 121. (7) Van Zee, R. J.; Li, S.; Weltner, W., Jr. J. Chem. Phys. 1993, 98, 4335. (8) Knight, L. B. Jr.; Petty, J. T. J. Chem. Phys. 1988, 88, 481. (9) Stwarts, C. A.; McGill, T. C.; Goddard III, W. A. Surf. Sci. 1981, 110, 400. (10) Stwarts, C. A.; Goddard III, W. A.; McGill, T. C. J. Vac. Sci. Technol. 1981, 19, 551. (11) Lamfried, W. H.; Strehlow, R. Phys. Status Solidi 1982, B110, K79. (12) Balasubramanian, K. J. Chem. Phys. 1987, 86, 3410; Erratum: J. Chem. Phys. 1990, 92, 2123. (13) Balasubramanian, K. Chem. ReV. 1990, 90, 93. (14) Balasubramanian, K. J. Mol. Spectrosc. 1990, 139, 405. (15) Meier, U.; Peyerimhoff, S. D.; Bruna, P. J.; Grein, F. J. Mol. Spectrosc. 1989, 134, 259.

Calculations on Electronic States of GaAs (16) Lou, L.; Wang, L.; Chibante, L. P. F.; Laaksonen, R. T.; Nordlander, P.; Smalley, R. E. J. Chem. Phys. 1991, 94, 8015. (17) Lou, L.; Nordlander, P.; Smalley, R. E. J. Chem. Phys. 1992, 97, 1858. (18) Hurley, M. M.; Pacios, L. F.; Christiansen, P. A.; Ross, R. B.; Ermler, W. C. J. Chem. Phys. 1986, 84, 6840. (19) Buenker R. J.; Peyerimhoff, S. D. Theor. Chim. Acta 1974, 35, 33. (20) Buenker R. J.; Peyerimhoff, S. D. Theor. Chim. Acta 1975, 39, 217. (21) Buenker, R. J. Int. J. Quantum Chem. 1986, 29, 435. (22) Buenker, R. J.; Peyerimhoff, S. D.; Butcher, W. Mol. Phys. 1978, 35, 771. (23) Buenker, R. J. In Proceedings of the Workshop on Quantum Chemistry and Molecular Physics; Burton, P., Ed.; University Wollon-

J. Phys. Chem. A, Vol. 102, No. 48, 1998 9883 gong: Wollongong, Australia; 1980; Studies in Physical and Theoretical Chemistry; Carbo´, R., Ed.; Elsevier: Amsterdam, 1981; Vol. 21 (Current Aspects of Quantum Chemistry). (24) Buenker, R. J.; Phillips, R. A. J. Mol. Struct. (THEOCHEM) 1985, 123, 291. (25) Davidson, E. R. In The World of Quantum Chemistry; Daudel, R., Pullman, B., Ed.; Reidel: Dordrecht, 1974. (26) Hirsch, G.; Bruna, P. J.; Peyerimhoff, S. D.; Buenker, R. J. Chem. Phys. Lett. 1977, 52, 442. (27) Das, K. K.; Liebermann, H. P.; Buenker, R. J.; Hirsch, G. J. Chem. Phys. 1995, 102, 4518. (28) Cooley, J. W. Math. Comput. 1961, 15, 363. (29) Moore, C. E. Atomic Energy LeVels; National Bureau of Standards: Washington, DC, 1971; Vol. 3.

Ab Initio Based Configuration Interaction Calculations on the Low

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